Currently I'm struggling with choosing between various analysis options, ranging from repeated measures design, to 4-way ANOVA to ANCOVA or to moderation analysis with the PROCESS macro of Hayes.

Some background information: My main research question is: To what extent can subgroup membership predict changes in X scores six months after participation in an intervention, and is this effect moderated by Y(controlled for Z.

I am not sure if I should work with a repeated measures design, or work with a change score of AUDIT (by calculating T1-T0). What I have read about this is that the difference scores ANOVA (1) tests whether the change or difference from T0 to T1 is equal acorss all groups, whereas ANCOVA (2) tests whether the T1 scores are equal across groups while controlling for their scores on T0. I've read that this is a small however potentially impactful distinction which got famous through Lord's paradox (https://m-clark.github.io/docs/lord/index.html / ANCOVA Versus CHANGE From Baseline in Nonrandomized Studies: The Difference: Multivariate Behavioral Research: Vol 48, No 6 (tandfonline.com)). It's been stated that if your groups are randomly assigned experimental groups, both methods are equivalent and you can choose whichever you prefer. If they are naturally occuring groups the literature indeed suggests using the difference scores method.

Since the subgroups I'm working with are latent classes that indeed 'naturally' occur, I am wondering if I should indeed go with change scores (despite its downfalls that has been written about in the literature, e.g. addition of measurement errors etc).

What is important to keep in mind is that my data (in both options, so taking AUDIT change or AUDIT_T1 as DV) has been violating assumptions of normality ánd homogeneity throughout, and I am not sure how to best deal with that in my current situation. The macro's of Hayes moderation PROCESS tool seemed like a good solution, but it depends on question 1 whether I can use that (because i need to use de AUDIT change score for that).

All in all, I am unsure how to proceed. Thank you in advance for thinking along.